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Theorem List for Metamath Proof Explorer - 17501-17600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxkohmeo 17501* The Exponential Law for topological spaces. The "currying" function  F is a homeomorphism on function spaces when  J and  K are exponentiable spaces (by xkococn 17349, it is sufficient to assume that  J ,  K are locally compact to ensure exponentiability). (Contributed by Mario Carneiro, 13-Apr-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  K  e.  (TopOn `  Y )
 )   &    |-  F  =  ( f  e.  ( ( J 
 tX  K )  Cn  L )  |->  ( x  e.  X  |->  ( y  e.  Y  |->  ( x f y ) ) ) )   &    |-  ( ph  ->  J  e. 𝑛Locally 
 Comp )   &    |-  ( ph  ->  K  e. 𝑛Locally 
 Comp )   &    |-  ( ph  ->  L  e.  Top )   =>    |-  ( ph  ->  F  e.  ( ( L 
 ^ k o  ( J  tX  K )
 )  Homeo  ( ( L 
 ^ k o  K )  ^ k o  J ) ) )
 
Theoremqtopf1 17502 If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F : X -1-1-> Y )   =>    |-  ( ph  ->  F  e.  ( J  Homeo  ( J qTop 
 F ) ) )
 
Theoremqtophmeo 17503* If two functions on a base topology 
J make the same identifications in order to create quotient spaces  J qTop  F and  J qTop  G, then not only are  J qTop  F and  J qTop  G homeomorphic, but there is a unique homeomorhism that makes the diagram commute. (Contributed by Mario Carneiro, 24-Mar-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F : X -onto-> Y )   &    |-  ( ph  ->  G : X -onto-> Y )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X )
 )  ->  ( ( F `  x )  =  ( F `  y
 ) 
 <->  ( G `  x )  =  ( G `  y ) ) )   =>    |-  ( ph  ->  E! f  e.  ( ( J qTop  F )  Homeo  ( J qTop  G ) ) G  =  ( f  o.  F ) )
 
Theoremt0kq 17504* A topological space is T0 iff the quotient map is a homeomorphism onto the space's Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )   =>    |-  ( J  e.  (TopOn `  X )  ->  ( J  e.  Kol2  <->  F  e.  ( J  Homeo  (KQ `  J ) ) ) )
 
Theoremkqhmph 17505 A topological space is T0 iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Kol2  <->  J  ~=  (KQ `  J )
 )
 
Theoremist1-5lem 17506 Lemma for ist1-5 17508 and similar theorems. If  A is a topological property which implies T0, such as T1 or T2, the property can be "decomposed" into T0 and a non-T0 version of property  A (which is defined as stating that the Kolmogorov quotient of the space has property  A). For example, if  A is T1, then the theorem states that a space is T1 iff it is T0 and its Kolmogorov quotient is T1 (we call this property R0). (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  A  ->  J  e.  Kol2 )   &    |-  ( J  ~=  (KQ `  J )  ->  ( J  e.  A  ->  (KQ `  J )  e.  A )
 )   &    |-  ( (KQ `  J )  ~=  J  ->  (
 (KQ `  J )  e.  A  ->  J  e.  A ) )   =>    |-  ( J  e.  A 
 <->  ( J  e.  Kol2  /\  (KQ `  J )  e.  A ) )
 
Theoremt1r0 17507 A T1 space is R0. That is, the Kolmogorov quotient of a T1 space is also T1 (because they are homeomorphic). (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Fre  ->  (KQ `  J )  e. 
 Fre )
 
Theoremist1-5 17508 A topological space is T1 iff it is both T0 and R0. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Fre  <->  ( J  e.  Kol2  /\  (KQ `  J )  e.  Fre ) )
 
Theoremishaus3 17509 A topological space is Hausdorff iff it is both T0 and R1 (where R1 means that any two topologically distinct points are separated by neighborhoods). (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( J  e.  Haus  <->  ( J  e.  Kol2  /\  (KQ `  J )  e.  Haus ) )
 
Theoremnrmreg 17510 A normal T1 space is regular Hausdorff. In other words, a T4 space is T3 . One can get away with slightly weaker assumptions; see nrmr0reg 17435. (Contributed by Mario Carneiro, 25-Aug-2015.)
 |-  ( ( J  e.  Nrm  /\  J  e.  Fre )  ->  J  e.  Reg )
 
Theoremreghaus 17511 A regular T0 space is Hausdorff. In other words, a T3 space is T2 . A regular Hausdorff or T0 space is also known as a T3 space. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( J  e.  Reg  ->  ( J  e.  Haus  <->  J  e.  Kol2 )
 )
 
Theoremnrmhaus 17512 A T1 normal space is Hausdorff. A Hausdorff or T1 normal space is also known as a T4 space. (Contributed by Mario Carneiro, 24-Aug-2015.)
 |-  ( J  e.  Nrm  ->  ( J  e.  Haus  <->  J  e.  Fre ) )
 
11.2  Filters and filter bases
 
11.2.1  Filter Bases
 
Syntaxcfbas 17513 Extend class definition to include the class of filter bases.
 class  fBas
 
Syntaxcfg 17514 Extend class definition to include the filter generating function.
 class  filGen
 
Definitiondf-fbas 17515* Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
 |- 
 fBas  =  ( w  e.  _V  |->  { x  e.  ~P ~P w  |  ( x  =/=  (/)  /\  (/)  e/  x  /\  A. y  e.  x  A. z  e.  x  ( x  i^i  ~P (
 y  i^i  z )
 )  =/=  (/) ) }
 )
 
Definitiondf-fg 17516* Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
 |-  filGen  =  ( w  e. 
 _V ,  x  e.  ( fBas `  w )  |->  { y  e.  ~P w  |  ( x  i^i  ~P y )  =/=  (/) } )
 
Theoremelmptrab 17517* Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  F  =  ( x  e.  D  |->  { y  e.  B  |  ph } )   &    |-  (
 ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  ps ) )   &    |-  ( x  =  X  ->  B  =  C )   &    |-  ( x  e.  D  ->  B  e.  V )   =>    |-  ( Y  e.  ( F `  X )  <-> 
 ( X  e.  D  /\  Y  e.  C  /\  ps ) )
 
Theoremelmptrab2 17518* Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  F  =  ( x  e.  _V  |->  { y  e.  B  |  ph } )   &    |-  (
 ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  ps ) )   &    |-  ( x  =  X  ->  B  =  C )   &    |-  B  e.  V   &    |-  ( Y  e.  C  ->  X  e.  W )   =>    |-  ( Y  e.  ( F `  X )  <->  ( Y  e.  C  /\  ps ) )
 
Theoremisfbas 17519* The predicate " F is a filter base." Note that some authors require filter bases to be closed under pairwise intersections, but that is not necessary under our definition. One advantage of this definition is that tails in a directed set form a filter base under our meaning. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( B  e.  A  ->  ( F  e.  ( fBas `  B )  <->  ( F  C_  ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P ( x  i^i  y ) )  =/=  (/) ) ) ) )
 
Theoremfbasne0 17520 There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( F  e.  ( fBas `  B )  ->  F  =/=  (/) )
 
Theorem0nelfb 17521 No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( F  e.  ( fBas `  B )  ->  -.  (/)  e.  F )
 
Theoremfbsspw 17522 A filter base on a set is a subset of the power set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( fBas `  B )  ->  F  C_  ~P B )
 
Theoremfbelss 17523 An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( fBas `  B )  /\  X  e.  F ) 
 ->  X  C_  B )
 
Theoremfbdmn0 17524 The domain of a filter base is nonempty. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( fBas `  B )  ->  B  =/=  (/) )
 
Theoremisfbas2 17525* The predicate " F is a filter base." (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( B  e.  A  ->  ( F  e.  ( fBas `  B )  <->  ( F  C_  ~P B  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  E. z  e.  F  z  C_  ( x  i^i  y
 ) ) ) ) )
 
Theoremfbasssin 17526* A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.)
 |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. x  e.  F  x  C_  ( A  i^i  B ) )
 
Theoremfbssfi 17527* A filter base contains subsets of its finite intersections. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  ( fi
 `  F ) ) 
 ->  E. x  e.  F  x  C_  A )
 
Theoremfbssint 17528* A filter base contains subsets of its finite intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( fBas `  B )  /\  A  C_  F  /\  A  e.  Fin )  ->  E. x  e.  F  x  C_  |^| A )
 
Theoremfbncp 17529 A filter base does not contain complements of its elements. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  A  e.  F ) 
 ->  -.  ( B  \  A )  e.  F )
 
Theoremfbun 17530* A necessary and sufficient condition for the union of two filter bases to also be a filter base. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X ) )  ->  ( ( F  u.  G )  e.  ( fBas `  X )  <->  A. x  e.  F  A. y  e.  G  E. z  e.  ( F  u.  G ) z  C_  ( x  i^i  y ) ) )
 
Theoremfbfinnfr 17531 No filter base containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( fBas `  B )  /\  S  e.  F  /\  S  e.  Fin )  ->  |^| F  =/=  (/) )
 
Theoremopnfbas 17532* The collection of open supersets of a nonempty set in a topology is a neighborhoods of the set, one of the motivations for the filter concept. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { x  e.  J  |  S  C_  x }  e.  ( fBas `  X ) )
 
Theoremtrfbas2 17533 Conditions for the trace of a filter base  F to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( ( F  e.  ( fBas `  Y )  /\  A  C_  Y )  ->  ( ( Ft  A )  e.  ( fBas `  A ) 
 <->  -.  (/)  e.  ( Ft  A ) ) )
 
Theoremtrfbas 17534* Conditions for the trace of a filter base  F to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  ( ( F  e.  ( fBas `  Y )  /\  A  C_  Y )  ->  ( ( Ft  A )  e.  ( fBas `  A ) 
 <-> 
 A. v  e.  F  ( v  i^i  A )  =/=  (/) ) )
 
11.2.2  Filters
 
Syntaxcfil 17535 Extend class notation with the set of filters on a set.
 class  Fil
 
Definitiondf-fil 17536* The set of filters on a set. Definition 1 (axioms FI, FIIa, FIIb, FIII) of [BourbakiTop1] p. I.36. Filters are used to define the concept of limit in the general case. They are a generalization of the idea of neighborhoods. Suppose you are in  RR. With neighborhoods you can express the idea of a variable that tends to a specific number but you can't express the idea of a variable that tends to infinity. Filters relax the "axioms" of neighborhoods and then succeed in expressing the idea of something that tends to infinity. Filters were invented by Cartan in 1937 and made famous by Bourbaki in his treatise. A notion similar to the notion of filter is the concept of net invented by Moore and Smith in 1922. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |- 
 Fil  =  ( z  e.  _V  |->  { f  e.  ( fBas `  z )  | 
 A. x  e.  ~P  z ( ( f  i^i  ~P x )  =/=  (/)  ->  x  e.  f ) } )
 
Theoremisfil 17537* The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  <->  ( F  e.  ( fBas `  X )  /\  A. x  e.  ~P  X ( ( F  i^i  ~P x )  =/=  (/)  ->  x  e.  F ) ) )
 
Theoremfilfbas 17538 A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  F  e.  ( fBas `  X ) )
 
Theorem0nelfil 17539 The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  -.  (/)  e.  F )
 
Theoremfileln0 17540 An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F ) 
 ->  A  =/=  (/) )
 
Theoremfilsspw 17541 A filter is a subset of the power set of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  F  C_  ~P X )
 
Theoremfilelss 17542 An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F ) 
 ->  A  C_  X )
 
Theoremfilss 17543 A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  ( A  e.  F  /\  B  C_  X  /\  A  C_  B ) ) 
 ->  B  e.  F )
 
Theoremfilin 17544 A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  i^i  B )  e.  F )
 
Theoremfiltop 17545 The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  X  e.  F )
 
Theoremisfil2 17546* Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  <->  ( ( F 
 C_  ~P X  /\  -.  (/) 
 e.  F  /\  X  e.  F )  /\  A. x  e.  ~P  X ( E. y  e.  F  y  C_  x  ->  x  e.  F )  /\  A. x  e.  F  A. y  e.  F  ( x  i^i  y )  e.  F ) )
 
Theoremisfildlem 17547* Lemma for isfild 17548. (Contributed by Mario Carneiro, 1-Dec-2013.)
 |-  ( ph  ->  ( x  e.  F  <->  ( x  C_  A  /\  ps ) ) )   &    |-  ( ph  ->  A  e.  _V )   =>    |-  ( ph  ->  ( B  e.  F  <->  ( B  C_  A  /\  [. B  /  x ].
 ps ) ) )
 
Theoremisfild 17548* Sufficient condition for a set of the form  { x  e.  ~P A  |  ph } to be a filter. (Contributed by Mario Carneiro, 1-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ph  ->  ( x  e.  F  <->  ( x  C_  A  /\  ps ) ) )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  [. A  /  x ].
 ps )   &    |-  ( ph  ->  -.  [. (/)  /  x ]. ps )   &    |-  ( ( ph  /\  y  C_  A  /\  z  C_  y )  ->  ( [. z  /  x ].
 ps  ->  [. y  /  x ].
 ps ) )   &    |-  (
 ( ph  /\  y  C_  A  /\  z  C_  A )  ->  ( ( [. y  /  x ]. ps  /\  [. z  /  x ].
 ps )  ->  [. (
 y  i^i  z )  /  x ]. ps )
 )   =>    |-  ( ph  ->  F  e.  ( Fil `  A ) )
 
Theoremfilfi 17549 A filter is closed under taking intersections. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( fi `  F )  =  F )
 
Theoremfilinn0 17550 The intersection of two elements of a filter can't be empty. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  i^i  B )  =/=  (/) )
 
Theoremfilintn0 17551 A filter has the finite intersection property. Remark below definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 20-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  ( A  C_  F  /\  A  =/=  (/)  /\  A  e.  Fin ) )  ->  |^| A  =/=  (/) )
 
Theoremfiln0 17552 The empty set is not a filter. Remark below def. 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 30-Oct-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  F  =/=  (/) )
 
Theoreminfil 17553 The intersection of two filters is a filter. Use fiint 7129 to extend this property to the intersection of a finite set of filters. Paragraph 3 of [BourbakiTop1] p. I.36. (Contributed by FL, 17-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  G  e.  ( Fil `  X ) )  ->  ( F  i^i  G )  e.  ( Fil `  X ) )
 
Theoremsnfil 17554 A singleton is a filter. Example 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( A  e.  B  /\  A  =/=  (/) )  ->  { A }  e.  ( Fil `  A ) )
 
Theoremfbasweak 17555 A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  F  C_  ~P Y  /\  Y  e.  V ) 
 ->  F  e.  ( fBas `  Y ) )
 
Theoremsnfbas 17556 Condition for a singleton to be a filter base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( A  C_  B  /\  A  =/=  (/)  /\  B  e.  V )  ->  { A }  e.  ( fBas `  B ) )
 
Theoremfsubbas 17557 A condition for a set to generate a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( X  e.  V  ->  ( ( fi `  A )  e.  ( fBas `  X )  <->  ( A  C_  ~P X  /\  A  =/=  (/)  /\  -.  (/)  e.  ( fi
 `  A ) ) ) )
 
Theoremfbasfip 17558 A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  -.  (/)  e.  ( fi
 `  F ) )
 
Theoremfbunfip 17559* A helpful lemma for showing that certain sets generate filters. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  Y ) )  ->  ( -.  (/)  e.  ( fi
 `  ( F  u.  G ) )  <->  A. x  e.  F  A. y  e.  G  ( x  i^i  y )  =/=  (/) ) )
 
Theoremfgval 17560* The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  ( X filGen F )  =  { x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) } )
 
Theoremelfg 17561* A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  ( A  e.  ( X filGen F )  <->  ( A  C_  X  /\  E. x  e.  F  x  C_  A ) ) )
 
Theoremssfg 17562 A filter base is a subset of its generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  F  C_  ( X filGen F ) )
 
Theoremfgss 17563 A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X )  /\  F  C_  G )  ->  ( X filGen F )  C_  ( X filGen G ) )
 
Theoremfgss2 17564* A condition for a filter to be finer than another involving their filter bases. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X ) )  ->  ( ( X filGen F )  C_  ( X filGen G )  <->  A. x  e.  F  E. y  e.  G  y  C_  x ) )
 
Theoremfgfil 17565 A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( X filGen F )  =  F )
 
Theoremelfilss 17566* An element belongs to a filter iff any element below it does. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  C_  X )  ->  ( A  e.  F  <->  E. t  e.  F  t 
 C_  A ) )
 
Theoremfilfinnfr 17567 No filter containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  S  e.  F  /\  S  e.  Fin )  ->  |^| F  =/=  (/) )
 
Theoremfgcl 17568 A generated filter is a filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( fBas `  X )  ->  ( X filGen F )  e.  ( Fil `  X ) )
 
Theoremfgabs 17569 Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( F  e.  ( fBas `  Y )  /\  Y  C_  X )  ->  ( X filGen ( Y
 filGen F ) )  =  ( X filGen F ) )
 
Theoremneifil 17570 The neighborhoods of a non empty set is a filter. Example 2 of [BourbakiTop1] p. I.36. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  S  C_  X  /\  S  =/= 
 (/) )  ->  (
 ( nei `  J ) `  S )  e.  ( Fil `  X ) )
 
Theoremfilunibas 17571 Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  U. F  =  X )
 
Theoremfilunirn 17572 Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  U. ran  Fil  <->  F  e.  ( Fil `  U. F ) )
 
Theoremfilcon 17573 A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( Fil `  X )  ->  ( F  u.  { (/) } )  e.  Con )
 
Theoremfbasrn 17574* Given a filter on a domain, produce a filter on the range. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
 |-  C  =  ran  (  x  e.  B  |->  ( F
 " x ) )   =>    |-  ( ( B  e.  ( fBas `  X )  /\  F : X --> Y  /\  Y  e.  V )  ->  C  e.  ( fBas `  Y ) )
 
Theoremfiluni 17575* The union of a nonempty set of filters with a common base and closed under pairwise union is a filter. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( F  C_  ( Fil `  X )  /\  F  =/=  (/)  /\  A. f  e.  F  A. g  e.  F  ( f  u.  g )  e.  F )  ->  U. F  e.  ( Fil `  X ) )
 
Theoremtrfil1 17576 Conditions for the trace of a filter  L to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  A  =  U. ( Lt  A ) )
 
Theoremtrfil2 17577* Conditions for the trace of a filter  L to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  ( ( Lt  A )  e.  ( Fil `  A ) 
 <-> 
 A. v  e.  L  ( v  i^i  A )  =/=  (/) ) )
 
Theoremtrfil3 17578 Conditions for the trace of a filter  L to be a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( L  e.  ( Fil `  Y )  /\  A  C_  Y )  ->  ( ( Lt  A )  e.  ( Fil `  A ) 
 <->  -.  ( Y  \  A )  e.  L ) )
 
Theoremtrfilss 17579 If  A is a member of the filter, then the filter truncated to  A is a subset of the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F ) 
 ->  ( Ft  A )  C_  F )
 
Theoremfgtr 17580 If  A is a member of the filter, then truncating  F to  A and regenerating the behavior outside  A using 
filGen recovers the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( F  e.  ( Fil `  X )  /\  A  e.  F ) 
 ->  ( X filGen ( Ft  A ) )  =  F )
 
Theoremtrfg 17581 The trace operation and the 
filGen operation are inverses to one another in some sense, with  filGen growing the base set and ↾t shrinking it. See fgtr 17580 for the converse cancellation law. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  ( ( F  e.  ( Fil `  A )  /\  A  C_  X  /\  X  e.  V )  ->  ( ( X filGen F )t  A )  =  F )
 
Theoremtrnei 17582 The trace, over a set  A, of the filter of the neighborhoods of a point  P is a filter iff  P belongs to the closure of  A. (This is trfil2 17577 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  Y )  /\  A  C_  Y  /\  P  e.  Y )  ->  ( P  e.  ( ( cls `  J ) `  A )  <->  ( ( ( nei `  J ) `  { P } )t  A )  e.  ( Fil `  A ) ) )
 
Theoremcfinfil 17583* Relative complements of the finite parts of an infinite set is a filter. When  A  =  NN the set of the relative complements is called Frechet's filter and is used to define the concept of limit of a sequence. (Contributed by FL, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( X  e.  V  /\  A  C_  X  /\  -.  A  e.  Fin )  ->  { x  e. 
 ~P X  |  ( A  \  x )  e.  Fin }  e.  ( Fil `  X )
 )
 
Theoremcsdfil 17584* The set of all elements whose complement is dominated by the base set is a filter. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( ( X  e.  dom  card  /\  om  ~<_  X ) 
 ->  { x  e.  ~P X  |  ( X  \  x )  ~<  X }  e.  ( Fil `  X ) )
 
Theoremsupfil 17585* The supersets of a nonempty set which are also subsets of a given base set form a filter. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
 |-  ( ( A  e.  V  /\  B  C_  A  /\  B  =/=  (/) )  ->  { x  e.  ~P A  |  B  C_  x }  e.  ( Fil `  A ) )
 
Theoremzfbas 17586 The set of upper integer sets is a filter base on  ZZ, which corresponds to convergence of sequences on  ZZ. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |- 
 ran  ZZ>=  e.  ( fBas `  ZZ )
 
Theoremuzrest 17587 The restriction of the set of upper integers to an upper integer set is the set of upper integers based at a point above the cutoff. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  ( ran  ZZ>=t  Z )  =  (
 ZZ>= " Z ) )
 
Theoremuzfbas 17588 The set of upper integer sets based at a point in a fixed upper integer set like  NN is a filter base on  NN, which corresponds to convergence of sequences on 
NN. (Contributed by Mario Carneiro, 13-Oct-2015.)
 |-  Z  =  ( ZZ>= `  M )   =>    |-  ( M  e.  ZZ  ->  ( ZZ>= " Z )  e.  ( fBas `  Z )
 )
 
11.2.3  Ultrafilters
 
Syntaxcufil 17589 Extend class notation with the ultrafilters-on-a-set function.
 class  UFil
 
Syntaxcufl 17590 Extend class notation with the ultrafilter lemma.
 class UFL
 
Definitiondf-ufil 17591* Define the set of ultrafilters on a set. An ultrafilter is a filter that gives a definite result for every subset. (Contributed by Jeff Hankins, 30-Nov-2009.)
 |- 
 UFil  =  ( g  e.  _V  |->  { f  e.  ( Fil `  g )  | 
 A. x  e.  ~P  g ( x  e.  f  \/  ( g 
 \  x )  e.  f ) } )
 
Definitiondf-ufl 17592* Define the class of base sets for which the ultrafilter lemma filssufil 17602 holds. (Contributed by Mario Carneiro, 26-Aug-2015.)
 |- UFL 
 =  { x  |  A. f  e.  ( Fil `  x ) E. g  e.  ( UFil `  x ) f  C_  g }
 
Theoremisufil 17593* The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( F  e.  ( UFil `  X )  <->  ( F  e.  ( Fil `  X )  /\  A. x  e.  ~P  X ( x  e.  F  \/  ( X 
 \  x )  e.  F ) ) )
 
Theoremufilfil 17594 An ultrafilter is a filter. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( F  e.  ( UFil `  X )  ->  F  e.  ( Fil `  X ) )
 
Theoremufilss 17595 For any subset of the base set of an ultrafilter, either the set is in the ultrafilter or the complement is. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( S  e.  F  \/  ( X  \  S )  e.  F )
 )
 
Theoremufilb 17596 The complement is in an ultrafilter iff the set is not. (Contributed by Mario Carneiro, 11-Dec-2013.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  S  C_  X )  ->  ( -.  S  e.  F 
 <->  ( X  \  S )  e.  F )
 )
 
Theoremufilmax 17597 Any filter finer than an ultrafilter is actually equal to it. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  G  e.  ( Fil `  X )  /\  F  C_  G )  ->  F  =  G )
 
Theoremisufil2 17598* The maximal property of an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
 |-  ( F  e.  ( UFil `  X )  <->  ( F  e.  ( Fil `  X )  /\  A. f  e.  ( Fil `  X ) ( F  C_  f  ->  F  =  f ) ) )
 
Theoremufprim 17599 An ultrafilter is a prime filter. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Mario Carneiro, 2-Aug-2015.)
 |-  ( ( F  e.  ( UFil `  X )  /\  A  C_  X  /\  B  C_  X )  ->  ( ( A  e.  F  \/  B  e.  F ) 
 <->  ( A  u.  B )  e.  F )
 )
 
Theoremtrufil 17600 Conditions for the trace of an ultrafilter  L to be an ultrafilter. (Contributed by Mario Carneiro, 27-Aug-2015.)
 |-  ( ( L  e.  ( UFil `  Y )  /\  A  C_  Y )  ->  ( ( Lt  A )  e.  ( UFil `  A ) 
 <->  A  e.  L ) )
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